PomeCorollary 14.19.29.AEA7DCEDBA8B Every bounded $\omega $-plurisubharmonic solution of $\omega _ u^ n=f\, \mathrm{d}V$ is continuous.
Corollary 14.19.29AEA7DCEDBA8B
Proof. Since $u$ is bounded, the density $\mathrm{e}^{-u}f$ is nonnegative, nonzero and belongs to $L^ p$. By theorem 14.19.3, applied with $\lambda =1$ to this density, there is a continuous solution $r$ of
\[ \omega _ r^ n=\mathrm{e}^{r-u}f\, \mathrm{d}V. \]
The bounded potential $u$ solves the same exponential equation. Applying lemma 14.19.28 with $f_ a=f_ b=\mathrm{e}^{-u}f$ in both orders gives $u=r$, so $u$ is continuous. $\square$
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